When Carol meets Mr Miller for the first time after the big party, he's hung over and playing a bass. It is obvious that he's not actually playing it. Although we hear different notes being played, his left hand never moves and his right hand doesn't always match the rhythm.
When Carol takes Jerry to the beach, at the end of the scene Jerry has the blue towel in his hand as he lies back into his seat. As Carol lies back next to him into her seat, the blue towel is now draped over the back of both their seats.
When "Rebel" (Edie Adams) testifies before the Ad Council, she says "Jerry" (Rock Hudson) considers the top of the Chrysler Building "Inspiration Point", because "it looks down on Madison Ave." The Chrysler Building looks down on Lexington Ave and because of buildings in the way, Madison Ave, two blocks away, is not clearly visible.
At one point in the film, Jerry and Carol talk with an aquarium behind them, which holds a fish that is eaten by another as the scene ends. The "aquarium" is clearly just a short loop of film being played forwards and backwards in succession - the fish repeats the same two motions over and over until it is eaten.